Reconstruction of Function Fields

Abstract

For k an algebraic closure of the finite field $$\mathbb{F}_p$$ , l prime distinct from p and X a surface over k, we prove that the field of rational functions k(X) can be recovered from the maximal pro-l-quotient $${\mathcal{G}}_{K}$$ of its absolute Galois group – in fact already from the second central descending series quotient of $${\mathcal{G}}_{K}$$ .